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Calculus. I: The first derivative of pseudoisotopy theory. (English) Zbl 0741.57021
A \(k\)-connected map \(Y\to X\) of manifolds induces a \((k-2)\)-connected map \({\mathcal P}(Y)\to {\mathcal P}(X)\) of the associated stable smooth pseudoisotopy spaces. The main result of the paper describes the \((2k- 3)\)-homotopy type of \(fiber({\mathcal P}(Y)\to {\mathcal P}(X))\). Let \(\Lambda(X)=Maps(S^ 1,X)\) denote the free loop space and \(\Lambda(Y\to X)=Y\times_ X\Lambda(X)\). The inclusion of constant loops \(X\to\Lambda(X)\) induces an inclusion \(Y\to \Lambda(Y\to X)\). The \((2k- 3)\)-homotopy type of \(fiber({\mathcal P}(Y)\to{\mathcal P}(X))\) is the same as that one of \(fiber(\Omega^ 2Q(\Lambda(Y\to X)/Y)\to \Omega^ 2Q(\Lambda(X)/X))\) with \(Q=\Omega^ \infty\Sigma^ \infty\). As a corollary the author determines the \((2k-3)\)-homotopy type of \(fiber(A(Y)\to A(X))\), Waldhausen’s relative \(K\)-theory of topological spaces. Using the theory of calculus of functors the proof reduces to a geometric analysis of the map \(P(N)\to P(M)\) of the unstable pseudoisotopy spaces induced by the inclusion \(N\subset M\), where \(M\) is obtained from \(N\) by attaching a handle. The calculus of functors, developed by the author in the past decade, is a theory particularly suited for stable range calculations and has had remarkable applications. The paper starts with a concise treatment of the differentiation and the derivative of homotopy functors (higher derivatives and the powerful theory of analytic functors are deferred to a later paper). The derivative of \(X\mapsto Q(Map(K,X)_ +)\), \(K\) a finite complex, is determined in Section 2. This example is of special interest in view of the connection of \(Map(S^ 1,X)\) to algebraic \(K\)-theory. The main result is phrased as calculation of the derivative of the functor \(X\mapsto {\mathcal P}(X)\), which determines the \((2k-3)\)-homotopy type of \(fiber({\mathcal P}(Y)\to {\mathcal P}(X))\). Its proof constitutes Section 3.

MSC:
57R52 Isotopy in differential topology
19D10 Algebraic \(K\)-theory of spaces
55P65 Homotopy functors in algebraic topology
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